Factor Using Both Sum and Product
Factoring a monic quadratic into two linear factors requires numbers matching both the linear coefficient’s sum and the constant term’s product.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Factor x²−5x+6. Candidate answers: (A) (x+2)(x+3) (B) (x−2)(x−3) (C) (x−5)² (D) (x−6)(x−1).
- Independent check inputs: Factor x²−7x+12. Candidate answers: (A) (x+3)(x+4) (B) (x−12)(x−1) (C) (x−3)(x−4) (D) (x−7)².
Learn the idea
Quadratic equations can be solved by revealing factors, making a square, or undoing a square. Each method exposes structure, and all valid roots must be retained and checked in the original equation. Factoring a monic quadratic into two linear factors requires numbers matching both the linear coefficient’s sum and the constant term’s product. Signs must satisfy both conditions.
Here is the method on the supplied teaching case. The constant term requires two numbers whose product is 6. Their sum must be −5; −2 and −3 meet both conditions.
Matching the product alone is insufficient. Multiply the proposed factors to recover the middle coefficient and constant; a sign error can preserve one condition while violating the other.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Factor x²−5x+6. Candidate answers: (A) (x+2)(x+3) (B) (x−2)(x−3) (C) (x−5)² (D) (x−6)(x−1). Solution: The constant term requires two numbers whose product is 6. Their sum must be −5; −2 and −3 meet both conditions. The final answer is (x−2)(x−3). Matching the product alone is insufficient. Multiply the proposed factors to recover the middle coefficient and constant; a sign error can preserve one condition while violating the other.
Check your reasoning
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Helpful tips
- Factoring a monic quadratic into two linear factors requires numbers matching both the linear coefficient’s sum and the constant term’s product.
- Matching the product alone is insufficient.
See the answer and explanation
(x−3)(x−4)
The constant term requires two numbers whose product is 12. Their sum must be −7; −3 and −4 meet both conditions.
Take it further
Changed-input task: Factor x²−19x+90. Candidate answers: (A) (x+9)(x+10) (B) (x−19)² (C) (x−9)(x−10) (D) (x−90)(x−1). Show the mathematical reason, not only the selected answer. Expected result: (x−9)(x−10). The constant term requires two numbers whose product is 90. Their sum must be −19; −9 and −10 meet both conditions.
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